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“I certainly understand and am puzzled in some ways, but the joint committee’s basic charge is to find in 10 weeks anywhere from $1.2 trillion to $1.5 trillion in savings over the next 10 years,” Durbin said. “In 10 weeks, these 12 Members of the House and Senate are to come together and reach an agreement. It’s a daunting task.”

Senate Minority Whip Jon Kyl (R-Ariz.), who participated in the May and June bipartisan talks with Vice President Joseph Biden, took a teasing tone with reporters Tuesday in declining to name a short list of candidates.

“A. Be a glutton for punishment; B. Have a real good bladder; and C. Be able to spend some time in August at this job, which none of us want to do,” Kyl quipped about the qualifications necessary to be a member of the panel.

In terms of who actually will get the call, lawmakers and aides are keeping mum. However, there seemed to be a consensus that appointing members of the former gang of six, for example, would be politically impossible. There could be some easy picks for each leader, however. McConnell could choose a Member such as Sen. Rob Portman (R-Ohio), a freshman who has already proved himself as a valuable counsel on budget matters to leaders. Portman also served as House Budget chairman and director of the Office of Management and Budget under President George W. Bush.

Picking gang of six member Sen. Tom Coburn (R-Okla.) would be tricky for McConnell, however, because Coburn has attracted the ire of conservatives who feel he has given too much ground on increasing revenues.

On the House side, Democratic aides note that Assistant Minority Leader James Clyburn (S.C.) and Caucus Vice Chairman Xavier Becerra (Calif.) would be consensus picks that appease minority groups and policy wonks. Rep. Chris Van Hollen (Md.), ranking member on the Budget Committee and a frequent point man for Pelosi on fiscal matters, could also be selected. And Rep. Robert Andrews (N.J.), an eloquent party spokesman and Pelosi favorite, could be an underdog choice.

Moderates have also been angling for a spot, even beginning their campaign before the deal was finalized. Several members of the New Democrat and Blue Dog coalitions sent a letter to Boehner and Pelosi last week saying, “We expect to see Members appointed who are committed to the balanced principles that are reflected in the proposals developed by Simpson-Bowles and the Gang of Six.”

Reps. Jim Himes (Conn.), Allyson Schwartz (Pa.) and Jim Cooper (Tenn.), who signed the letter, were named by several Democratic aides as potential choices from the party’s more moderate wing.

No amount of speculation from reporters could prompt Pelosi into discussing her potential picks at a press conference Tuesday, and Boehner has been equally mum and likely won’t give any signals as to his picks before they’re announced.

“It’s obviously an important decision, and the Speaker will be consulting with the entire leadership team and Members of our Conference as we move forward,” Boehner spokesman Michael Steel said.

Asked during a CNBC interview Tuesday about whether he wanted to serve on the new panel, House Budget Chairman Paul Ryan (R-Wis.) said, “I’ll serve if the Speaker wants me to serve.”